Cyclic trelles coded modulation

ABSTRACT

A universal method of trellis encoding signals mapped according to any signal constellation format involves constructing an encoder output table and a state transition table. The encoder output table defines the output symbol of an encoder given the input symbol and the present state of the encoder, while the state transition table defines the next state of the encoder given the present state of the encoder and the input applied to the encoder. The output table and the next state table are constructed with the objective of providing maximal distances between the branches of the trellis diagram without any regards for the shift register implementation of the code. Cyclic trellis-coded modulation is an example of such codes without feed-forward or feed-back shift register implementations, and with equal or better performance than &#34;optimal&#34; shift register trellis codes with 16 states or less. The cyclic trellis codes for both AWGN and Rayleigh fading applications can be constructed for any signal constellation without resorting to exhaustive searches.

This application is a divisional of U.S. patent application Ser. No.08/344,111, filed Nov. 23, 1994 U.S. Pat. No. 5,675,590.

FIELD OF THE INVENTION

The present invention relates to digital communication systems, and, inparticular, to forward error correction through trellis codedmodulation.

BRIEF DESCRIPTION OF THE RELATED ART

In recent years, much of the research and development in thecommunications industry has been concentrated in the area of digitalsignal transmission. As is well known in the art, digital signaltransmission typically involves transmission of data with a carrierfrequency. The carrier frequency is modulated by data so that afrequency bandwidth is occupied by the transmitted signal. The growingdemand for access to data and communication services has placed asignificant strain on the available bandwidth. Moreover, there is anever increasing demand for increased data communication rates for thepurpose of decreasing the data transmission time. An increase of therate of the data typically results in an increased bandwidthrequirement, placing a further strain upon the available bandwidth fortransmission of signals.

In an effort to increase the data rates without sacrificing theavailable bandwidth, a number of increasingly sophisticated codedmodulation schemes have been developed. For example, quadratureamplitude modulation (QAM) employs both amplitude and phase modulationin order to encode more data within a given frequency bandwidth. Anothermodulation technique involves multiple phase shift keying (MPSK) toincrease data capacity within a given bandwidth. These high levelmodulation schemes are very sensitive to channel impairments. That is,the information encoded by means of such techniques is often lost duringtransmission due to noise, Rayleigh fading and other factors which areintroduced over the communication medium.

In order to compensate for the increased sensitivity of these high levelmodulation schemes, various forward error correction coding techniquesare employed One such error coding technique is trellis codedmodulation. Trellis coded modulation is desirable since it combinesmodulation and error coding operations to provide effective errorcontrol coding without sacrificing power and bandwidth efficiency.Furthermore, it has been shown that trellis coded modulation schemesperform significantly better than their uncoded equivalents with thesame power and bandwidth efficiency. Trellis codes have been developedfor many of the high-level, high-rate modulation schemes, includingwell-known 8-PSK modulation and Square 16 QAM modulation. However,designers of past systems have not considered providing a technique oftrellis coding which applies to any phase and/or amplitude modulationscheme, as well as codes having various constraint lengths, whileproviding optimal or near optimal error performance.

Typically, a new set of "optimal" trellis codes must be foundindividually for each modulation scheme. The "optimal" trellis codes aretypically found through algorithms that search all the possible trelliscode structures that have simple feedback of feed-forward shift registerimplementations. Even a small change in system parameters, such as codeconstraint length, requires a search for an entirely new set of trelliscodes.

SUMMARY OF THE INVENTION

The present invention provides a system and method for trellis-codedmodulation and demodulation of phase and/or amplitude modulated signalscomplying with varying signal constellations and constraint lengths,while yielding optimum or near optimum error performance. The cyclictrellis encoding method of the present invention can generate a familyof trellis codes whose performance is better than or equal to so called"optimal" codes generated by other techniques. Generally, the cyclictrellis codes do not have a feed-forward or feed-back shift registerimplementation.

A method of cyclic trellis encoding a data sequence within an encoder isdisclosed. The data sequence is to be mapped according to apredetermined modulation scheme having an associated signalconstellation. The signal constellation has defined coordinate pointscorresponding to phase and amplitude characteristics corresponding tooutput symbols from the encoder. The method comprises the step ofdefining an output table of output symbols. The output table has presentstate rows and input symbol columns. The output symbols are determinedas a function of symbols input to the encoder and a present state of theencoder. The method of the present invention further comprises the stepof defining the output table, which further comprises the substeps ofassigning each of the output symbols to the points of the signalconstellation; partitioning the points of the signal constellation intoa first subset of output symbols and a second subset of output symbols;loading even ones of the present state rows with output symbols from thefirst subset; and loading odd ones of the present state rows with outputsymbols from the second subset. The method further comprises the step ofdefining a next-state table of next states for the encoder. Thenext-state table has present state rows and input symbol columns,wherein the next states are defined as a function of symbols input tothe encoder and a present state of the encoder. The method furthercomprises the step of defining the next-state look-up table, whichfurther comprises the substeps of loading first ones of the presentstate rows with next states of the encoder until at least one of thefirst present state rows is full and all of the next state values havebeen used; and loading other ones of the present state rows with nextstates that are cyclicly shifted from the next states in each of thefirst ones of the present state rows until all of the present state rowsare filled. The method of the present invention further comprises thestep of implementing the output and next-state tables within the encoderso that output symbols from the encoder are determined by input symbolsto the encoder and the present state of the encoder in accordance withthe output table, and transitions from the present state of the encoderto the next state of the encoder are performed in accordance with thenext-state table. Finally, the method comprises the step of mapping theoutput symbols into signals having phase and amplitude characteristicscorresponding to points on the signal constellation.

In a preferred embodiment of the present invention, the coordinatepoints of the signal constellation are assigned output symbols accordingto natural mapping techniques when the predominant channel interferenceis Additive White Gaussian Noise.

In another preferred embodiment the coordinate points of the signalconstellation are assigned to output symbols according to Gray codingtechniques when the predominant channel interference is Rayleigh fading.

In another embodiment of the present invention, a method of cyclictrellis encoding an input data sequence with an encoder is disclosed.The input data sequence is to be mapped according to a predeterminedmodulation scheme having an associated signal constellation. The signalconstellation has defined coordinate points corresponding to phase andamplitude characteristics of output symbols from the encoder. Theencoder receives n inputs, corresponding to 2^(n) input values, andoutputs n+1 outputs corresponding to 2^(n+1) output values. The encoderhas 2^(k) possible states. The method of the present invention comprisesthe steps of defining an output table having 2^(k) present state rowsand 2^(n) input symbol columns, wherein the output symbols from theencoder are determined as a function of input symbols to the encoder anda present state of the encoder; defining the output table which furthercomprises the substeps of assigning values to the points of the signalconstellation, where the values correspond to the output symbols;partitioning the signal constellation into a first subset of 2^(n)output symbols and a second subset of 2^(n) output symbols, where thefirst subset and second subset are symmetric; loading even ones of thepresent state rows with values corresponding to output symbols from thefirst subset; and loading odd ones of the present state rows with valuescorresponding to output symbols from the second subset. The method ofthe present invention further comprises the step of defining anext-state table of a plurality of next states, where the next-statetable has 2^(k) present state rows and 2^(n) input symbol columns. Thenext state of the encoder is determined as a function of the inputsymbols to the encoder and the present state of the encoder. The step ofdefining the next-state table further comprises the substeps of dividingthe next states into 2^(k-n) subsets wherein each subset has 2^(n) nextstates; and loading a first one of the present state rows with nextstates from a first one of the subsets, a second one of the presentstate rows with the next states from a second one of the subsets, andcontinuing this loading until the 2^(k-n) th present state row is loadedwith the next states from the 2^(k-n) th one of the subsets. The methodof the present invention further comprises the steps of implementing theoutput and next-state tables within the encoder so that output symbolsfrom the encoder are determined as a function of input symbols to theencoder and the present state of the encoder in accordance with theoutput table, and transitions from the present state of the encoder tothe next state of the encoder are in accordance with the next-statetable; and mapping the output symbols from the encoder into signalshaving phase and amplitude characteristics corresponding to respectiveoutput symbol points on the signal constellation.

In a preferred embodiment of the invention, the coordinate points of thesignal constellation are assigned output symbols according to naturalmapping techniques when the predominant channel interference is AdditiveWhite Gaussian Noise.

In another preferred embodiment of the invention, the coordinate pointsof the signal constellation are assigned output symbols according toGray coding techniques when the predominant channel interference isRayleigh fading.

Under another aspect, the present invention provides for a trellisencoder which trellis encodes input data signals, wherein the input datasignals are mapped according to a modulation scheme such that a signalconstellation defined by the modulation scheme cannot be set partitionedsuch that each level of set partitioning results in a substantiallyincreased minimum Euclidean distance between points of the signalconstellation.

Another embodiment of the present invention calls for a transmitter fora trellis-coded, multi-level modulation communication system. Thetransmitter comprises a cyclic trellis encoder which receives a sequenceof data input symbols and outputs a sequence of encoded output symbols.The cyclic trellis encoder has a set of present states partitioned intosubsets. The encoder comprises a state transition table containing aplurality of next state values for the encoder. The next state valuesare defined based upon the present state of the encoder and the inputsymbol. The next state values are assigned to each of the present statesubsets such that the next state values for any present state subset areshifted cyclicly for successive members of the any present state subset.A present state memory element connects to the state transition look-uptable and temporarily stores a next state value output by the statetransition look-up table. An encoder output look-up table connects tothe present state memory element which selects an output symbol basedupon the present state of the encoder and the presently received one ofthe input symbols. The present states and the output values arepartitioned into two subsets so that the output look-up table outputs asymbol which belongs to a first output subset when in one of the presentstate subsets and outputs a symbol which belongs to a second outputsubset when in the other of the present state subsets. A signal mapperconnects to the output look-up table, and maps outputs of the encoderoutput look-up table into encoded output signals from two symmetricsignal constellations. Finally, a transmitter circuit transmits theencoded output signals over a communications medium.

In a preferred embodiment, the signal mapper maps according to Graycoding techniques. In another preferred embodiment, the signal mappermaps according to natural mapping techniques.

Under another aspect, the apparatus of the present invention comprises areceiver for a trellis-coded multi-level modulation communicationsystem. The receiver comprises a trellis decoder which receives abaseband signal. The trellis decoder comprises a means forreconstructing a trellis structure defined by a state transition look-uptable wherein a set of present states is partitioned into subsets havingsuccessive members, and next state values assigned to each of thepresent state subsets are shifted cyclicly for successive members of thepresent state subset. The trellis decoder further comprises a means fordetermining input and output symbols associated with branches of thetrellis structure as defined by an encoder output look-up table. Thepresent states and the output values are partitioned into two subsets sothat the output look-up table outputs a symbol which belongs to a firstoutput subset when in one of the present state subsets, and outputs asymbol which belongs to a second output subset when in the other of thepresent state subsets. The decoder also includes a calculation circuitwhich determines the Euclidean distances between points on aphase/amplitude coordinate system corresponding to the received signalsand points of a phase/amplitude signal constellation corresponding tosignals associated with branches on the trellis structure. Finally, thetrellis decoder comprises a comparator circuit which selects the mostlikely path of the received signal on the trellis structure on the basisof the determined Euclidean distances.

In a further embodiment of the present invention, a method is disclosedof forward error correction coding for a data signal mapped according toa given signal constellation. The method comprises the steps of defininga family of convolutional codes. The family of convolutional codes ischaracterized in that the family of codes are not capable of beinggenerated by means of a feed-forward or feed-back shift registerimplementation. The step of defining further comprises the substeps ofestablishing a next-state value corresponding to eachpresent-state/input-value pair; establishing an output valuecorresponding to each present-state/input-value pair; receiving an inputdata symbol corresponding to an input value; providing an output valuein response to the reception of the input value, wherein the outputvalue is determined by the input value and the present-state value, andthe present state value is determined by the previous next state value;generating an output data symbol, wherein the output symbol isdetermined by the output value; and encoding a data signal to correspondto the output symbol as determined by a signal mapping scheme.

In accordance with a still further aspect of the present invention, amethod of encoding signals mapped according to any signal constellationformat comprises the steps of dividing the signal constellation pointsinto two symmetrical sets of symbol points, and cyclic trellis encodingdata to be mapped according to the signal constellation.

Under a further aspect, the present invention comprises a data encoder,the data encoder comprises a cyclic trellis encoder and a transmittercoupled to the cyclic trellis encoder.

Under a yet further aspect, the present invention comprises a trellisencoder that may be implemented as a state machine or a look-up tablememory or in software, but which cannot be implemented as a shiftregister. In one embodiment, the cyclic trellis encoder is such as notto allow a feed-forward shift register implementation.According to afurther aspect, the cyclic trellis encoder is such as not to allow afeed-back shift register implementation.

In yet a further embodiment, the present invention is a data encoder foruse in data communication application, wherein a plurality of input datavalues are trellis encoded to form output data. The data encodercomprises an input and an encoder circuit coupled to the input. Theencoder circuit has a plurality of present states, and is responsive tothe plurality of input data values on the input to transition to a nextstate. The next states correspond to each input which is cycliclyshifted for different ones of the present states. The encoder furthercomprises an output coupled to the encoder circuit. The output isresponsive to the encoder circuit to generate output data.

A trellis encoder constructed in accordance with the teachings of thepresent invention can be adapted to trellis encode data to be mapped forsignal constellations. The signal constellations can be set partitionedsuch that each level of set partitioning results in substantiallyincreased minimum Euclidean distance between points of the signalconstellation. The trellis encoder can also be adapted to trellis encodedata to be mapped for signal constellations which cannot be setpartitioned such that each level of set partitioning results insubstantially increased minimum Euclidean distance between points of thesignal constellation.

BRIEF DESCRIPTION OF THE DRAWINGS

FIGS. 1A and 1B are graphical representations of signal constellationscorresponding to Square 16 QAM and 16 Star QAM signals, respectively.

FIG. 2A is an exemplary convolutional encoder circuit.

FIG. 2B is a state table describing the operation of the convolutionalencoder of FIG. 2A.

FIG. 3 is a trellis state transition diagram representing the operationof the convolutional encoder circuit of FIG. 2A.

FIG. 4A-4C are 4-PSK signal constellations which illustrate the methodused to determine trellis path probabilities according to Euclideandistances along a trellis diagram represented in FIG. 4D.

FIG. 5 is a trellis set partitioning tree for a 4-PSK signalconstellation.

FIG. 6 is a schematic representation of a trellis coding tree whichgraphically depicts the signal constellation space for a conventionallyencoded Square 16 QAM signal constellation at each level of trelliscoding.

FIG. 7 is a schematic representation of a trellis coding tree whichgraphically depicts the signal constellation space for a 16 Star QAMsignal constellation at each level of trellis coding in accordance withthe method of the present invention.

FIG. 8 is a simplified block diagram of a wireless communication systemwhich employs cyclic trellis error encoding.

FIG. 9 depicts a single level set partitioning of an 8-PSK signalconstellation coded by natural coding techniques.

FIG. 10 depicts a 16 state, rate 2/3 encoder output look-up table forany 8 level modulation scheme in AWGN environment which defines theencoder output value given the present state of the encoder and theinput applied to the encoder.

FIG. 11 depicts a single level set partitioning of an 8-PSK signalconstellation coded by Gray coding techniques.

FIG. 12 depicts a 16 state, rate 2/3 encoder output look-up table forany 8 level modulation scheme in Rayleigh fading environment whichdefines the encoder output value given the present state of the encoderand the input applied to the encoder.

FIG. 13 depicts a 16 state transition table for any 8 level modulationscheme which defines the next state of the encoder given the presentstate of the encoder and the input to the encoder.

FIG. 14 depicts a single level set partitioning of a 16 Star QAM signalconstellation coded by natural coding.

FIG. 15 depicts a 16 state, rate 3/4 encoder output look-up table forany 16 level modulation scheme in AWGN environment which defines theencoder output value given the present state of the encoder and theinput applied to the encoder.

FIG. 16 depicts a single level set partitioning of a 16 Star QAM signalconstellation coded by Gray coding techniques.

FIG. 17 depicts an encoder output look-up table for any 16 levelmodulation scheme in Rayleigh fading environment which defines theencoder output value given the present state of the encoder and theinput applied to the encoder.

FIG. 18 depicts a state transition table for any 16 level modulationscheme which defines the next state of the encoder given the presentstate of the encoder and the input to the encoder.

FIG. 19 depicts an encoder output look-up table in generalized formwhich defines the encoder output value given the present state of theencoder and the input applied to the encoder for any signalconstellation and any code constraint length, for applcations in AWGNenvironment.

FIGS. 20A-20C depict a state transition table in generalized form whichdefines the next state of the encoder given the present state of theencoder and the input to the encoder for any signal constellation andany code constraint length.

FIG. 21 is a schematic block diagram which shows the main structural andfunctional elements of a cyclic trellis encoder constructed inaccordance with the teachings of the present invention.

FIG. 22 depicts a state transition table for any 8 level modulationscheme which may be used to avoid catastrophic codes.

DETAILED DESCRIPTION OF THE INVENTION

FIGS. 1A and 1B graphically depict signal constellations for a Square 16QAM scheme and a 16 Star QAM scheme, respectively. The signalconstellations depicted in FIGS. 1A and 1B are merely for exemplarypurposes to illustrate different signal constellation formats such asmay be commonly used in the art. The signal constellations depicted inFIGS. 1A and 1B are represented in polar coordinate form wherein eachpoint of the signal constellations represents the determinate phase andthe amplitude of an information symbol. For example, a point labeled "Q"in FIG. 1B corresponds to a signal having an amplitude defined as 1 anda phase of 45°, while a point labeled "P" in FIG. 1B corresponds to asignal having an amplitude defined as 2 which has a phase of 90°. Forthis particular 16 Star QAM constellation, the inner ring of signalpoints are amplitude value 1 points and the outer ring of signal pointsare amplitude value 2 points.

Signal constellations are a convenient way to graphically depict thebinary data encoded by means of various phase and amplitude modulationschemes. For example, as shown in FIGS. 1A and lB, there is a 4-bitbinary word ("symbol") associated with each point on the 16-point signalconstellations. This means that a detector is configured to assign aspecified data word for a detected signal having a given amplitude andphase. Thus, for example, in the 16 Star QAM constellation of FIG. 1B,when a detector (or decoder) detects a signal which has an amplitudeclosest to 1, and which has a phase closest to 45°, the detector willassign the data word 0011 (decimal 3) to that signal. Similarly, when adetector detects a signal which has an amplitude closest to 2, and whichhas a phase closest to 90°, the detector will assign the data word 0101(decimal 5) to that signal. The data assignments in FIGS. 1A and 1B aremerely exemplary. Other assignments are possible, as will be apparentfrom the discussion below.

To minimize the effects of additive white Gaussian noise (AWGN) as wellas the effects of Rayleigh fading and other channel impairments, one ormore error encoding techniques are used in order to provide for accuratetransmission and detection of data, especially when very high levelmodulation schemes are employed.

TRELLIS-CODED MODULATION

Trellis-coded modulation is a forward error correction coding techniquewhich is also well known in the art. Trellis codes are convolutionalcodes that are designed and optimized according to a specific modulationscheme. A convolutional encoder encodes information symbols based uponthe present input symbol and the state of the encoder. The present stateof the encoder is determined by the symbols which previously entered theencoder. That is, the encoded symbol is a function of the present inputsymbol and also symbols that entered the encoder before the presentinput symbol. Thus, a convolutional encoder has memory.

Convolutional codes are typically implemented by shift registers andsummers. The next state and the output of the encoder are functions ofthe present state of the register or look-up table (i.e., the value ofthe bits presently stored within the register or look-up table memory),and the input to the register or look-up table.

FIG. 2A and the accompanying table 230 shown in FIG. 2B illustrate anexemplary embodiment of a convolutional encoder 200 implemented by meansof shift registers, and the corresponding state table. The encoder 200is simply shown here in order to illustrate the operation andimplementation of a convolutional encoder, and is not to be construed asan implementation of the trellis encoder used in accordance with thepresent invention. The encoder 200 includes shift register memory units205, 210, 215, as well as summers 220, 225. A one-bit input is encodedinto a two-bit output to provide rate 1/2 encoding.

Assuming an initial state of 000 (i.e., the register units 205, 210, 215contain bit values of 0, 0, 0, respectively), and an input value of 0,the next state of the encoder 200 is 000 (a zero bit value shifts inwhile a zero value shifts out). Consequently, the value of the two bitsat the output is 00. This is represented in the first line of the statetable 230 in FIG. 2B. Note, however, that the present and next statecolumns only indicate two-bit values since the last state bit is alwaysshifted out and is not significant in determining the next state. Thus,when moving from state to state, the encoder 200 can be considered tohave four possible present states and four possible next states, eachtwo-bit values. As another example, assume the encoder 200 to be in thepresent state 10 (i.e., the first two registers contain 1,0). An inputof 1 will move the encoder 200 to a next state of 11 (i.e., the firsttwo registers contain 1,1) and generate an output of 01 (decimal 1).This process is repeated as each successive bit enters the encoder 200so that a state diagram can be constructed which shows the possiblestate transitions of the encoder 200 with the accompanying input andoutput values which correspond to those transitions.

FIG. 3 is a state transition diagram which indicates the possible statetransitions of the encoder 200 of FIG. 2, along with the input andoutput values corresponding to the possible transitions. Because thestate transition diagram resembles a trellis in form, such diagrams areoften called trellis diagrams, hence the name "trellis coding." Each doton the trellis diagram of FIG. 3 represents a state of the encoder 200.Dots in the same horizontal row correspond to the same state atdifferent times. Dots in the same vertical column represent differentstates at the same time (i.e., within the duration of the same symbol).Branches between the dots represent possible state transition paths.Thus, for example, there is a branch between the state 01 and the state00 which indicates that, given the appropriate input, the encoder 200could go from state 01 to state 00. Since there is no branch betweenstates 01 and 11, nor is there a branch between the states 01 and 01, itis not possible for the encoder 200 to go from state 01 to either of thestates 11 or 01 within one symbol duration.

The number pair along each of the branches depicted in FIG. 3 indicatethe input, output! values which correspond to a given branch. The firstnumber represents the input which causes the transition, while thesecond number represents the output value resultant upon thistransition.

As seen from the trellis diagram of FIG. 3, the possible statetransitions for the encoder 200 are the same for each successive symbol.Thus, the same pattern repeats over and over again for each symbolduration.

As an example, assume the encoder 200 begins in the state 0 (binary 00),represented by a dot 300 in FIG. 3. Upon application of an input value 1to the encoder 200, the encoder 200 goes from state 0 to state 2 (binary10), represented by a dot 320, via a path 310. Upon completion of thetransition, the encoder 200 outputs a value 3 (binary 11). If the valueof the next bit applied to the input is 0, then the encoder 200transitions from state 2 to state 1, represented by a dot 340, via apath 330, while the output of the encoder 200 assumes a value of 2.Finally, upon application of an input bit of 0, the encoder 200 movesfrom the state 1 to the state 0, represented by a dot 360, via a path350. Upon entering the state 0, the encoder 200 outputs a value 3. Thus,in the foregoing example, input bits 1-0-0 are encoded by the encoder200 into output bits 11-10-11, or 3-2-3 in decimal. At the same time,the encoder 200 has transitioned from the state 0 to the state 2, to thestate 1, and back to the state 0.

Maximum Likelihood Viterbi Decoding

As further explained below, convolutional encoding (and Viterbidecoding) provides for a reduced number of detected errors at thereceiver. Consider again the trellis diagram of FIG. 3. For example,assume that a three-bit data stream 1-0-0 is properly encoded as11-10-11 by the encoder 200 as described above. Also suppose that thereceiver detects the transmitted signal erroneously as 11-11-11. Inorder to determine what the original transmitted data is, the decoderperforms a maximum likelihood decision based upon the possible statetransition paths which the encoder 200 might have taken. Since theencoder is typically set to state 0 at initialization, the decoderassumes that the detected sequence of data bits began in state 0. Thedecoder then examines all of the paths which began at state 0 andterminate at a state three symbols later as depicted in FIG. 3 for thepurpose of illustration. For instance, for an ending point at the state0, at the point 360, there are two possible paths which the encoder mayhave taken: the path 310, 330, 350, or the path 370, 380, 390. Ofcourse, all the other paths of three symbol duration are also examinedto determine the likelihood that the detected bit sequence followedthese possible paths, but for the sake of simplicity of illustration,only the paths from state 0 to state 0 are considered here.

In order to identify the most likely path, the decoder determines theprobability that the detected data sequence was produced by the firstpath (e.g., the path 310, 330, 350), the probability that the detecteddata sequence was generated by the second path (e.g., the path 370, 380,390), and so on until a probability has been calculated for eachpossible path. The path having the highest probability is then selectedas the actual path according to either hard or soft decision methodsdescribed in greater detail below.

Typically, trellis decoding techniques calculate path probabilitiesbased upon either Hamming or Euclidean distances between the detectedsignal and the signals generated by the possible trellis paths. Inaccordance with the teachings of the present invention, Euclideandistances are used as the measure of path probability, as discussed ingreater detail below. However, in order to provide a clearerunderstanding of the method of determining the probability of a possibletrellis path, a brief discussion of Hamming distance is also provided,

Hamming Distance (Hard Decision Decoding)

Hamming distance is defined as the number of bits by which two binarysequences differ. For example, the hamming distance between the binarywords 110 and 101 is two, while the hamming distance between the binarywords 111 and 011 is one, etc. Based upon a Hamming distance evaluationof the possible paths, the probability that a given path has generated adetected data sequence can be determined as follows. Assuming, as statedabove, that the detected data sequence is 11-11-11 (with a proper datasequence of 11-10-11), and the possible paths are the paths 310, 330,350 and 370, 380, 390, the Hamming distance between the detected signal11-11-11 and the path 310, 330, 350 is 1. That is, because the path 310generates an output of 3 (11), the path 330 generates an output of 2(10), and the path 350 generates an output of 3 (11), the binarysequence generated by the path 310, 330, 350 is 11-10-11. This sequencediffers from the detected sequence 11-11-11 by a Hamming distance of 1.The Hamming distance between the detected signal 11-11-11 and the signalgenerated by the path 370, 380, 390 is 6 since the path 370, 380, 390results in an output binary sequence of 00-00-00. Thus, it is much morelikely that the detected sequence 11-11-11 was generated by the path310, 330, 350, than by the path 370, 380, 390. Therefore, it is morelikely that the sequence of input bits is 1-0-0.

Euclidean Distance (Soft Decision Decoding)

Another measure of the probability that a given path has generated abinary sequence is based upon Euclidean distance. Euclidean distance isthe length of a straight line between points on a signal constellation.In general, probability measures based upon Euclidean distances exhibitbetter accuracy than probability measures based upon Hamming distance.This is because probability measurements based upon Euclidean distancetake into account the received signal phase and amplitude informationwhich is discarded when using Hamming distance as a probability metric.For example, FIGS. 4A-4D illustrate a simple 4-PSK modulation signalconstellation having four defined points 400, 410, 420, 430 equidistantfrom the origin and corresponding to output values 00, 01, 10, and 11,respectively. Suppose a sequence of received data symbols are detectedto have phase and amplitude values which are represented by the vectorsr1-r3 in FIGS. 4A-4C. Using conventional Hamming decoding techniques,the vectors r1-r3 would simply be approximated as the data points 00,10, and 00, respectively, so that valuable phase and amplitudeinformation is lost about the actually detected signal sequence. Inaccordance with Euclidean techniques, however, the phase and amplitudeof the received signal are factored into the determination of the pathprobability.

As shown in FIG. 4D, the probability that the detected signal has beengenerated by the trellis path represented by the dashed line 450 is adecreasing function of the sum of the squares of the Euclidean distancesd01, d02, and d03 (depicted in FIGS. 4A-4C), while the probability thatthe detected signal has been generated by the trellis path representedby the dashed/dotted line 470 is a function of the sum of the squares ofthe Euclidean distances d31, d22, and d33. The greater the sum of thesquares of the Euclidian distances along a given path, the less likelythat path is to be the one which generated the detected signal sequence.In this manner, a more accurate estimation of the transmitted datasequence can be obtained.

It should be understood, of course, that as the number of points in thesignal constellation (i.e., the number of possible output values) andthe number of states in the trellis encoder increase, the number ofpossible trellis paths increases as well. Thus, for example, a rate 3/4trellis encoder which operates in conjunction with a 16 pointconstellation will have 8 possible branches merging into and divergingout of each state (represented by a point) on the trellis statetransition diagram. In these systems, the probability associated witheach path merging into a state point is determined. Once theseprobabilities have been compared, the path with the highest probabilityis determined and the corresponding data bits in that path are selectedas the decoded sequences.

Block and Symbol-By-Symbol Decoding Methods

The selection of a given path may be made in accordance with block orsymbol-by-symbol decision methods. In the case of a block decision, apredetermined number of received signals forming a set (e.g., 1,000symbols) are fed into the decoder. The decoder then starts with thefirst signal and constructs a trellis with associated metrics and pathhistories for the whole set of 1,000 symbols. The trellis transitionpath that is most probable is then selected as the path which generatedthe detected symbols. The data input which would have generated thispath is then determined as the decoded data sequence. Absent anyuncorrected errors, this data sequence should correspond to the datasequence fed into the encoder on the transmitter side of thecommunication system. The process is then repeated with the next blockof symbols, and so on.

For symbol-by-symbol decisions, a predetermined number of receivedsignals are fed into the decoder. For example, assume 25 signals are fedinto the decoder. Once the 25th symbol is entered, the trellis decoderdetermines what path was most probable. The input symbol which wouldhave generated the first branch of the most probable path is thenselected as the output of the decoder. The next (e.g., the 26th)received signal is then fed into the decoder and another determinationis made of the most probable path for the last 25 symbols (i.e.,excluding the first symbol). The input symbol which would have generatedthe first branch of the most probable path (i.e., the path for the mostrecently detected 25 symbols) is then selected as the next output of thedecoder. This procedure is carried on symbol-by-symbol in real time sothat only one symbol at a time is decoded for output as opposed to anentire block of data at a time.

Maximizing Euclidean Distance in Trellis Coding

Gottfried Ungerboeck, in a paper entitled "Channel Coding withMultilevel/Phase Signals," published January, 1982 in IEEE TRANSACTIONSON INFORMATION THEORY, Vol. IT-28, No. 1, and herein incorporated byreference, argued that error performance of convolutional codes could beimproved if designed by maximizing the Euclidean distances betweentrellis paths which merge into and out of the same state. This isaccomplished by tailoring the convolutional coding scheme to the signalconstellation of a given modulation technique so that the operations oferror coding and modulation are essentially combined.

Take as a simple example a 4-PSK signal constellation as shown in FIG.5. The possible outputs of the trellis encoder on the transmitter sideare represented as four points which are phase shifted from one anotherby phase differences of 90°. In any trellis coding scheme the possibleoutput values, as represented in the signal constellation, as well asthe states of the trellis decoder are both considered. In order toprovide the maximum distinction between encoded signals, so as to allowfor more accurate decoding, it is advantageous to assure thattransitions to and from the same state differ greatly in their outputvalues (in terms of their Euclidean distances). For example, the trellisdiagram of FIG. 3, which may, for example, describe state transitionsfor the 4-PSK signal constellation of FIG. 5, has the branches 370, 310diverging from the same state point 300. Note that the output value forthe state transition branch 310 is 3, and the output value for the statetransition branch 370 is 0. In accordance with the Ungerboeck teaching,these two output values differ by the maximum Euclidean distance (i.e.,a Euclidean distance of Δ=2 as represented in FIG. 5). In a similar way,state transitions resulting in the same output values are assigned astransitions between two different states. Note, for instance, that thetransition path 310 which results in an output value of 3 advances fromstate 00 to state 10, while a transition path 395 which also results inan output value of 3, advances from state 01 to state 00. The Ungerboeckmethod thus assures good discrimination between the encoded datasignals.

The most common method of trellis encoding in accordance withUngerboeck's teachings is set partitioning, of which a simple example isshown in FIG. 5. By partitioning the original 4-PSK signal into two setsof diametrically opposed 2-PSK signals based upon the state of thetrellis encoder, the maximum Euclidean distance can be maintainedbetween outputs merging into or diverging out of the same state. Suchset partitioning diagrams are commonly referred to as trellis codingtrees.

FIG. 6 graphically represents a trellis coding tree for set partitioninga more complicated Square 16 QAM signal constellation by the Ungerboeckmethod, which is well known in the art. As shown in FIG. 6, a complexsignal constellation is broken up into subsets. It is a requirement ofUngerboeck's set partitioning method that the minimum Euclideandistances measured between any of the points on the subsetconstellations exceed the minimum Euclidean distance between points onthe constellation from which the subsets are derived. Thus, for example,as shown in FIG. 6, the minimum Euclidean distance between any twopoints on the original constellation at the top of the trellis codingtree is less than the minimum Euclidean distance between any points ofthe constellation shown in subsets B₀ or B₁. In like fashion, theminimum Euclidean distances between any two points on the constellationsubsets C₀ and C₂ is greater than the minimum Euclidean distance betweenany two points in the subset B₀, and so on. As detailed above, anincreased minimum Euclidean distance between any two points in a signalconstellation insures that the probability of mistaking similar encoderoutput sequences is minimized. The error performance of the coded schemeis a function of the minimum Euclidean distance between any two givenpaths. To reduce the probability of error, the minimum Euclideandistance must be increased.

Difficulty of Set Partitioning Certain Signal Constellations

Unlike the Square 16 QAM signal constellation, the 16 Star QAM signalconstellation does not allow for division into subsets such that theminimum Euclidean distance between points increases significantly foreach level of partitioning. FIG. 7 illustrates the difficulty of setpartitioning the 16 Star QAM signal constellation. For the first levelof set partitioning there is no difficulty; the constellation dividessymmetrically, and the minimum Euclidean distance between the point onthe first subset is considerably greater than the minimum Euclideandistance between the points on the original constellation. However, atthe second level of set partitioning, the minimum Euclidean distancesare substantially the same as at the first level of set partitioning.Due to this characteristic of the 16 Star QAM signal constellation, ithas been thought that it is not possible to effectively trellis code a16 Star QAM signal using conventional Ungerboeck coding techniques.

The present invention includes a specially designed trellisencoder/decoder within a transmit/receive communications system whichsolves the problems associated with the set partitioning of the 16 StarQAM constellation. The encoder/decoder is constructed to encodeaccording to an inventive method called cyclic trellis coding which hasbeen found to work advantageously for signal constellations such as 16Star QAM or any arbitrary signal constellation. Cyclic trellis coding isdescribed in detail below.

CYCLIC TRELLIS CODING

Cyclic trellis coding in accordance with the present invention involvesa method of error encoding which may be adapted to the 16 Star QAMsignal constellation. In addition, cyclic trellis coding according tothe present invention can be applied to modulation schemes with anysignal constellation that can be partitioned into two symmetric subsets.Moreover, it has been verified that error performance of a system usingcyclic trellis coding, together with a 16 level modulation scheme, is asgood or better than systems employing the previously used Ungerboeckcodes for trellis encoders having up to 16 states.

The main criteria for the design of cyclic trellis codes is to maximizethe distance between the output signals on the branches of a trellismerging into or out of the same state, and to have a regular codestructure so that codes of varying constraint length and modulation canbe constructed without extensive searches. The cyclic trellis codes havevery predictable and regular distance profiles. Cyclic trellis codes arealso systematic codes. That is, the first n bits of the n+1-bit encodedoutput symbol are identical to the corresponding n-bit symbol input intothe encoder.

Simplified Cyclic Trellis Coding Communication System

FIG. 8 depicts a simplified block diagram showing the main functionalelements of a communication system 800 which employs cyclic trelliscoding in accordance with the teachings of the present invention. Aswill be appreciated by those of ordinary skill in the art, the system800 depicted in FIG. 8 is highly simplified and does not show blockinterleaving elements, synchronization insertion and detection elements,filtering elements, and other elements which are typically associatedwith digital communication systems. The system 800 includes a cyclictrellis encoder 810 which couples to a transmitter 830. The transmitter830 transmits radio frequency (RF) signals over an antenna 840. The RFsignals are received via a communication channel by an antenna 850 whichpasses the signals to a receiver 860. The receiver connects to a cyclictrellis decoder 870.

In operation, the cyclic trellis coding system 800 receives data whichis to be encoded. The data is then cyclicly encoded within the encoder810 according to the method described with reference to FIGS. 13-15below. Once the data has been encoded, the encoder 810 then maps theencoded data into the appropriate signal constellation by introducingthe appropriate phase and amplitude variations in the signal to encodethe signal according to a particular modulation scheme (e.g., 8-PSK, 16Star QAM, etc.). Once the signal is mapped, the signal is communicatedto the transmitter 830, which transmits the signal over thecommunication channel via the antenna 840. The antenna 850 receives thesignal; the receiver 860 detects the signal on the carrier and outputsthe detected signal at the baseband level. Finally, the cyclic trellisdecoder 870 decodes the received baseband signal and outputs datacorresponding to the data input at the transmitter side.

Method of Cyclic Trellis Encoding

To cyclic trellis encode a signal in accordance with the presentinvention, the signal constellation is first partitioned into two sets.This is the same as dividing the possible outputs of the encoder intotwo sets of output values. The sets into which the output symbols arepartitioned depend upon the particular mapping scheme. The mappingscheme which is used depends upon the particular application of thecommunication system. For applications involving mobile radiocommunications characterized by Rician or Rayleigh fading, the signalmapping, and hence the set partitioning, is performed according to Graycoding. For wireline applications, or any other applications, where theprimary channel impairment is that of AWGN, the signal mapping, andhence the set partitioning, is performed according to natural mapping.

The possible states of the encoder are also divided into a number ofsets. The output and state partitioning is most readily represented intable form. Because a trellis code is entirely determined by fivefactors (i.e., the input, the present state, the next state, the output,and the signal mapping) a trellis code can be wholly represented by twotables, and a signal mapper. The first table represents the encoderoutput as a function of the present state of the encoder and the inputapplied to the encoder. The second table represents the next state ofthe encoder as a function of the present state of the encoder and theinput applied to the encoder. The signal mapper determines thecoordinates of the output symbols in the signal constellation. For thesake of clarity, because the method of cyclic trellis coding is somewhatuniversal, the following description proceeds inductively with thepresentation of specific examples followed by a summary of the universalmethod of the present invention.

16 State Cyclic Trellis Code for 8-PSK Signals in AWGN

FIG. 9 depicts a one level partitioned 8-PSK signal constellation withnatural mapping for AWGN applications. As stated above, applicationswherein AWGN is the primary channel impairment require that the outputtable be defined in accordance with a naturally mapped signalconstellation. In natural mapping a reference point is chosen on thesignal constellation and it is assigned to symbol 0. The remainingsignals are then numbered consecutively around the constellation.

The tables shown in FIGS. 10 and 13 define the rate 2/3, 16-state cyclictrellis code for 8-PSK signals in AWGN. A generalized version of such anencoder is shown below, and described in detail with reference to FIG.21. FIG. 10 depicts an encoder output look-up table which defines theencoder output value given the present state of the encoder and theinput applied to the encoder. FIG. 13 depicts a next-state look-up tablewhich defines the next state of the encoder given the present state ofthe encoder and the input applied to the encoder. Since, in the presentexample, the encoder is a rate 2/3 encoder there are two input bits andthree output bits corresponding to four possible input values and eightpossible output values as depicted in FIG. 10. Thus, for example, whenan input of 2 (binary 10) is applied to a cyclic trellis encoderpresently in the state 9 (binary 1001), the encoder outputs a value of 5(binary 101). That is to say, the encoder encodes the output symbol suchthat the symbol is mapped onto the point having a value of 5 on thenatural-coded 8-PSK signal constellation. Once the symbol correspondingto a value of 5 is output from the encoder, the encoder moves from thestate 9 to the state 4, as represented in the table of FIG. 13.

As stated above, the cyclic trellis encoding method of the presentinvention is used to encode signals as defined by the tables shown inFIGS. 10 and 13. First, the possible output values of the encoder arepartitioned into two sets wherein the first set comprises all of theoutput values of the subset A (see FIG. 9), and the second set comprisesall of the output values of the subset B (FIG. 9). Thus, in accordancewith natural mapping, the first set comprises all of the even outputvalues and the second set comprises all of the odd output values. Foreach even present state value, an even output value is generated uponapplication of an input value, such that input values in ascending order(i.e., 0, 1, 2, 3) generate even output values in ascending order (i.e.,0, 2, 4, 6). In like fashion, for each odd present state value, an oddoutput value is generated upon application of an input value, such thatinput values in ascending order (i.e., 0, 1, 2, 3) generate odd outputvalues in ascending order (i.e., 1, 3, 5, 7). This form carries througheach of the present state values, as represented in FIG. 10.

16 State Cyclic Trellis Code for 8-PSK Signals in Rayleigh Fading

FIG. 11 depicts a one level partitioned 8-PSK signal constellation withGray-coded mapping for Rayleigh fading applications. As stated above,for wireless communications applications, the mapping scheme is that ofGray coding. As is well known in the art, Gray coding is a methodwhereby signal constellation points are assigned binary data values suchthat the Euclidean distances between points in adjacent signalconstellations correspond to a Hamming distance of one.

The tables shown in FIGS. 12 and 13 define the rate 2/3, 16-state cyclictrellis code for 8-PSK signals in Rayleigh fading. FIG. 12 depicts anencoder output look-up table which defines the encoder output valuegiven the present state of the encoder and the input applied to theencoder. Since, in the present example, the encoder is a rate 2/3encoder there are two input bits and three output bits corresponding tofour possible input values and eight possible output values as depictedin FIG. 12. Thus, for example, when an input of 2 (binary 10) is appliedto a cyclic trellis encoder presently in the state 9 (binary 1001), theencoder outputs a value of 4 (binary 100). That is to say, the encoderencodes the output symbol such that the symbol is mapped onto the pointhaving a value of 4 on the Gray-coded 8-PSK signal constellation. Oncethe symbol corresponding to a value of 4 is output from the encoder, theencoder moves from the state 9 to the state 4, as represented in thetable of FIG. 13.

As stated above, the cyclic trellis encoding method of the presentinvention is used to encode signals as defined by the tables shown inFIGS. 12 and 13. First, the possible output values of the encoder arepartitioned into two sets wherein the first set comprises all of theoutput values of the subset A (see FIG. 11), and the second setcomprises all of the output values of the subset B (FIG. 11). Thus, foreven present states (i.e., 0, 2, 4, . . . 14), in accordance withGray-coded mapping, an output value is generated upon application of aninput value such that input values in ascending order (i.e., 0, 1, 2, 3)generate the output values from the subset A in ascending order (i.e.,0, 3, 5, 6). In like fashion, for each odd present state value, anoutput value is generated upon application of an input value, such thatinput values in ascending order (i.e., 0, 1, 2, 3) generate outputvalues from the subset B in ascending order (i.e., 1, 2, 4, 7).

When the output value is generated by the encoder as defined by theoutput look-up table of either FIG. 10 or 12 (as called for by thespecific application), the encoder goes to the next state as defined bythe state transition or next-state look-up table of FIG. 13. Althoughthe form of the output look-up table is dependent upon the specificapplication (i.e., either AWGN or Rayleigh fading applications), thenext-state look-up table of FIG. 13 does not depend upon the predominantchannel impairment. The table of FIG. 13 is defined by filling up therows of the next-state table in ascending order until the lastnext-state value is placed. Thus, in the table of FIG. 13, the first rowis filled from 0 to 3, the second row is filled from 4 to 7, the thirdrow is filled from 8 to 11, and the fourth row is filled from 12 to 15.The entire pattern is then repeated with one modification: each row iscyclicly shifted by one place, so that the last value in the row becomesthe first, and the other values retain their order but are shifted tothe right by one space. Once all 16 states have been used (i.e., thenext four rows have been filled), the pattern repeats again so that thelast value is taken from the second set of four rows and is placed inthe first position while each of the other values is shifted over by oneplace. This procedure is repeated until all of the present state valuerows have been defined. Once the next-state look-up table has beendefined, the table can be implemented within an encoder, as described inmore detail below, by means of look-up table read only memories (ROMs),software, or other means as called for by the particular application.

16 State Cyclic Trellis Code for 16 Star QAM Signals in AWGN

FIG. 14 depicts a one level partitioned 16 Star QAM signal constellationwith natural mapping for AWGN applications. Output and next-state tablesto define a cyclic trellis encoder for coding of a 16 Star QAM signal inAWGN are depicted in FIGS. 15 and 18. The table shown in FIG. 15presents the output values as a function of the input and the presentstate values, while the table shown in FIG. 18 presents the next-statevalues as a function of the input and present state values.

In accordance with the teachings of the present invention, a rate 3/4cyclic trellis encoder for encoding a 16 Star QAM signal in AWGNpartitions the output values into two sets, as represented in the tableof FIG. 15. As is well known in the art, a rate 3/4 trellis encoderreceives 3 input lines (corresponding to 8 possible input values) andgenerates outputs on four lines (corresponding to 16 possible outputvalues). For this particular embodiment of the present invention, it hasbeen found that a 16 state encoder is advantageous, so that there are 16present state values and 16 next-state values. As shown in the table ofFIG. 15, each of the even present state values generates an output valuefrom the subset A (see FIG. 14) upon the application of an input value.Similarly, each of the odd present state values generates an outputvalue from the subset B (FIG. 14) upon the application of an inputvalue. Thus, in accordance with natural mapping, for each even presentstate value, an even output value is generated upon application of aninput value, such that input values in ascending order (i.e., 0, 1, 2, .. ., 6, 7) generate even output values in ascending order (i.e., 0, 2,4, . . ., 12, 14). In like fashion, for each odd present state value, anodd output value is generated upon application of an input value, suchthat input values in ascending order (i.e., 0, 1, 2, . . ., 6, 7)generate odd output values in ascending order (i.e., 1, 3, 5, . . ., 13,15). For example, if the trellis encoder is presently in state 7 (binary0111) and an input of 4 (binary 0100) is applied, the output valueproduced by the encoder will be 9 (binary 1001).

16 State Cyclic Trellis Code for 16 Star QAM Signals in Rayleigh Fading

FIG. 16 depicts a one level partitioned 16 Star QAM signal constellationwith Gray-coded mapping for Rayleigh fading applications. Output andnext-state tables to define a cyclic trellis encoder for coding of a 16Star QAM signal in Rayleigh fading are depicted in FIGS. 17 and 18. Thetable shown in FIG. 17 presents the output values as a function of theinput and the present state values.

In accordance with the teachings of the present invention, a rate 3/4cyclic trellis encoder for encoding a 16 Star QAM signal in Rayleighfading partitions the output values into two sets, as represented in thetable of FIG. 17. Each of the even present state values generates anoutput value from the subset A (see FIG. 17) upon the application of aninput value. Similarly, each of the odd present state values generatesan output value from the subset B (FIG. 17) upon the application of aninput value. Thus, in accordance with Gray-coded mapping, for each evenpresent state value, an output value from the subset A is generated uponapplication of an input value, such that input values in ascending order(i.e., 0, 1, 2, . . ., 6, 7) generate output values from the subset A inascending order (i.e., 0, 3, 5, 6, 9, 10, 12, 15). In like fashion, foreach odd present state value, an output value from the subset B isgenerated upon application of an input value, such that input values inascending order (i.e., 0, 1, 2, . . ., 6, 7) generate output values fromthe subset B in ascending order (i.e., 1, 2, 4, 7, 8, 11, 13, 14). Thus,for example, if the trellis encoder is presently in state 7 (binary0111) and an input of 4 (binary 0100) is applied, the output valueproduced by the encoder will be 8 (binary 1000).

The table of FIG. 18 represents the next state of a 3/4 cyclic trellisencoder as a function of the input and present state values. Althoughthe output table for the 16 Star QAM signal constellation encoderdepends upon whether the communication system is to be used in an AWGNenvironment or a Rayleigh fading environment, the form of the next-statelook-up table does not change due to the predominant channel impairment.As shown in FIG. 18, the next-state value is partitioned into multiplesets. That is, the first eight next-state values (0-7) are assigned, inorder, to the first present state value, while the next eight next-statevalues (8-15) are assigned, in order, to the second present state value.The first eight next-state values are then assigned again to the thirdpresent state value, but this time not in order from 0 to 7. Rather, theorder is 7, 0, 1, 2, 3, 4, 5, 6. Thus, the third next-state row is likethe first next-state row, but cyclicly shifted to the right. Thispattern is repeated between the fourth and second rows with the 15 beingshifted into the first position while the 14 is shifted into the lastposition of the row. Stated in general form, every next-state row takesthe last value from the row immediately preceding the previousnext-state row and places this value in the first position whileshifting the other values over by one position. This procedure isrepeated until an entire next-state table is defined. Generally, thecyclic trellis codes of the present invention do not have a feed-forwardor feed-back shift register implementation. However, as is well known inthe art, an encoder defined by the tables in FIGS. 15 or 17, and 18 maybe implemented as a look-up table, or some other input/output statemachine circuit.

Cyclic Trellis Coding For Any Signal Constellation

Specific examples detailing the method of cyclic trellis coding the8-PSK signal constellation and the 16 Star QAM signal constellation havebeen described above. The following description sets forth the generalmethod used for cyclic trellis encoding data mapped into any signalconstellation. As detailed in the examples above, the form of the outputlook-up table for any cyclic trellis coded signal constellation variesdepending upon the predominant channel impairment. Thus, if AWGN is thepredominant impairment, then natural coding of the signal constellationwill be used to determine the output look-up table, while if Rayleighfading is the predominant impairment, then Gray coding of the signalconstellation will be used to determine the output look-up table. Theform of the next-state look-up table does not vary with the predominantchannel impairment so that the form of the next-state look-up table willbe the same for either application.

Given a cyclic trellis encoder which receives n input bits at a time andencodes these into n+1 output bits at a time, there are 2^(n) possibleinput values and 2^(n+1) possible output values. Further, the number ofstates of the trellis encoder can be generally expressed as 2^(k)states, where k can be any positive integer. Thus, according to thisformulation, a cyclic trellis encoder can be represented by tables (likethose described above for 8-PSK and 16 Star QAM signal constellations)having 2^(n) input values, 2^(n+1) output values, and 2^(k) presentstate and next-state values. Therefore, the output table and the nextstate table have 2^(k) rows and 2^(n) columns.

The first step in defining a generalized encoder output table is topartition the set of 2^(n+1) output values into two sets. The elementsin each set will depend upon whether the table is defined for AWGNapplications or Rayleigh fading applications. The following descriptionis with reference to AWGN applications where natural mapping of thesignal constellation is employed. A brief description of the generalizedmethod of defining the output look-up table for Rayleigh fadingapplications is provided after the detailed description of the method ofdefining the output look-up table for AWGN applications.

For AWGN applications, the signal constellation is mapped according tonatural mapping. Once the signal constellation has been partitioned, inconjunction with natural mapping of the constellation, the first subsetcontains all of the even output values and the second subset containsall of the odd output values. The first set is then placed in ascendingorder (i.e., 0, 2, 4, . . ., 2^(n+1) -4, 2^(n+1) -2) in all of the evenpresent state rows. The second set is placed in ascending order (i.e.,1, 3, 5, . . ., 2^(n+1) -3, 2^(n+1) -1) in all of the odd present staterows. Such a generalized output table is represented in FIG. 19. Theoutput table shown in FIG. 19 may, for example, be implemented within acyclic trellis encoder by means of a ROM look-up table, or a softwarelook-up table, as will be described in greater detail below withreference to FIG. 21.

If the output look-up table is to be defined for Rayleigh fadingapplications, then the signal constellation is Gray coded. The signalconstellation is then partitioned into two symmetrical subsets, A and B.The elements of the output table for even present states are those fromsubset A in ascending order, while the elements of the output table forthe odd present states are those from the subset B in ascending order.

The form of the state transition table is the same for either AWGNapplications or Rayleigh fading applications. Defining a statetransition, or next-state table involves partitioning the set of 2^(k)present state values into m present state sets, where m equals 2^(k)/2^(n) (i.e., m=2^(k-n), which is the ratio between the number of statesand the number of input values) . For the present example, only the casewhere m is greater than or equal to one is explained. Thus, the firstpresent state set, S₀, contains the present state values {0, m, 2 m, . .., (2^(n) -1)m}, the second present state set, S₁, contains the presentstate values {1, m+1, 2 m+1, . . .,(2^(n) -1)m+1}, and so on until thelast set, S_(m-1), which contains the present state values {m-1, 2 m-1,. . ., (2^(n) -1)m+m-1}. For ease of illustration, FIGS. 20A-20Cillustrate state transition tables for each of the sets S₀ -S_(m-1). Itshould be understood, however, that the state transition tables depictedin FIGS. 20A-20C could be combined into one single state transitiontable, such as represented in FIGS. 10 and 15, which may be implementedwithin a cyclic trellis encoder by means of a software realized look-uptable, a ROM look-up table, or other input/output state circuitry.

In the table depicted in FIG. 20A, the rows of the state transitiontable correspond to the present state values in the first present stateset S₀. In the first row (i.e., the row corresponding to the presentstate 0), the first 2^(n) next-state values are assigned in ascendingorder from 0 to 2^(n) -1. In the second row of the table of FIG. 20A(i.e., the row corresponding to the present state value m), the samenext-state values are assigned for each input value. However, in row m,a cyclic shift has been performed so that the last value from row 0 isplaced in the first position of row m, while each of the remainingvalues are then placed in ascending order, from 0 to 2^(n) -2, for theremainder of row m. Another shift is performed for the next row in FIG.20A (i.e., the row corresponding to the present state value 2 m), sothat the next-state values in the row 2 m are 2^(n) -2, 2^(n) -1, 0, 1,2,. . ., 2^(n) -3. In this manner a cyclic shift is performed every rowof FIG. 20A (i.e., every mth present state value) until the last presentstate value in the set S₀ is reached.

A similar procedure is used to define the tables of FIGS. 20B and 20C.In the table depicted in FIG. 20B, the rows of the state transitiontable correspond to the present state values in the second present stateset S₁. In the first row (i.e., the row corresponding to the presentstate 1), the second 2^(n) next-state values are assigned in ascendingorder from 2^(n) to 2^(n+1) -1. In the second row of the table of FIG.20B (i.e., the row corresponding to the present state value m+1), thesame next-state values are assigned for each input value. However, inrow m+1, a cyclic shift has been performed so that the last value fromrow 1 is placed in the first position of row m+1, while each of theremaining values are then placed in ascending order, from 2^(n) to2^(n+1) -2, for the remainder of row m+1. Another shift is performed forthe next row in FIG. 20B (i.e., the row corresponding to the presentstate value 2m+1), so that the next-state values in the row 2^(m+) 1 are2^(n+1) -2, 2^(n+1) -1, 2^(n), 2^(n+1) +1, 2^(n+1) +2,. . ., 2^(n+1) -3.In this manner a cyclic shift is performed every row of FIG. 20B (i.e.,every mth present state value) until the last present state value in theset S₁, is reached.

FIG. 20C depicts a table which generally defines the state transitionsfor the ith set of present state values, S_(i). In the first row (i.e.,the row corresponding to the present state i), the i+1th 2^(n)next-state values are assigned in ascending order from 2^(n+i-1) to2^(n+i) -1. In the second row of the table of FIG. 20C (i.e., the rowcorresponding to the present state value m+i), the same next-statevalues are assigned for each input value. However, in row m+i, a cyclicshift has been performed so that the last value from row i is placed inthe first position of row m+i, while each of the remaining values arethen placed in ascending order, from 2^(n+i-1) to 2^(n+i) -2, for theremainder of row m+i. Another shift is performed for the next row inFIG. 20C (i.e., the row corresponding to the present state value 2 m+i),so that the next-state values in the row 2 m+i are 2^(n+i) -2, 2^(n+i)-1, 2^(n+i), 2^(n+i) +1, 2^(n+i) +2,. . ., 2^(n+i) -3. In this manner acyclic shift is performed every row of FIG. 20C (i.e., every mth presentstate value) until the last present state value in the set S₁ isreached.

The foregoing description of the construction of the state transitiontables of FIGS. 20A-20C has assumed that the ratio between the number ofstate values of the encoder and the number of input values of theencoder is greater than or equal to one (i.e., m>=1). If, however, thereare more possible input values than encoder states, the followingprocedure should be followed to cyclicly trellis encode the incomingdata signal.

The first present state row is filled by placing the next-state valuesin ascending order beginning at the first input value column until allof the next-state values have been placed. Since there are more inputvalues than next-state values, the next-state values will not fill theentire first row. Thus, the next-state values are again placed in thefirst row in ascending order. This process is repeated until the entirefirst row is filled. Because the number of next-state and input valuescan only be a power of two, the entire set of next-state values will fitan even number of times into the first row of the state transition tableso that there is no spill over into the next row. Once the first row hasbeen filled, the second row is defined by performing a cyclic shift ofthe first row. The third row is defined by performing a cyclic shift onthe second row, etc., until all of the rows have been filled within thestate transition look-up table.

According to the above procedure, an entire next-state table isconstructed which defines every next-state value as a function of theinput value and the present state value. According to this method theconstructed encoder output and next-state tables are general enough tobe applied to any signal constellation. As indicated above, the codesgenerated by the above-described method of cyclic trellis coding havebeen found not to have a feed-forward or feedback shift registerimplementation. The tables may be implemented within a cyclicconvolutional encoder as look-up tables, as described in detail below.

Properties of Cyclic Trellis Codes

It will be appreciated by those of ordinary skill in the art thatsignals coded according to the above-described technique have certainadvantageous properties which provide for optimal or near optimaldiscrimination amongst signals output from the cyclic trellis encoder810.

For instance, trellis state transition structures will have the minimumnumber of two branch paths for state transitions which originate out ofone state and merge back into the same state. For example, the 16 StarQAM signal which is cyclic trellis encoded will have only four statetransition paths which originate from the state zero and merge back intothe state zero within two transitions (i.e., the paths from state 0 tostate 0 and again to state 0; 0 to 2 back to 0; 0 to 4 back to 0; and 0to 6 back to 0).

Another advantageous property of cyclic trellis codes is that such codesprovide maximal or near maximal spacing between outputs on statetransition paths which originate from one state and merge into the samestate.

An additional advantageous property of cyclic trellis codes is that suchcodes can be constructed in a very short period of time for signalconstellations of virtually any form. Typically, a cyclic trellis codecan be constructed in a matter of minutes for most signal constellationswhile conventional simulation techniques for generating trellis codestake weeks or months by means of iterative computer searches. Thus, thecyclic trellis encoding technique of the present invention offers manysignificant advantages over the prior systems and methods which generateor employ trellis codes.

The Cyclic Trellis Encoder

FIG. 21 is a schematic block diagram which shows the main structuralelements of the general cyclic trellis encoder 810 constructed inaccordance with the teachings of the present invention. The encoder 810includes a state transition look-up table 1500, which may, for example,be fabricated from a ROM IC chip, or realized in software or otherinput/output state machine circuitry. A next state output of the statetransition look-up table 1500 connects to a memory element 1510 via aline 1505. The memory element 1510 may, in one embodiment, beimplemented as a series of D-flip flops. The output of the memoryelement 1510 connects to a present state input of the state transitionlook-up table 1500 via a line 1515 and to a present state input of anoutput look-up table 1520 via a line 1525. The output look-up table 1520connects to a signal mapping look-up table 1530 via a line 1535. Thestate transition look-up table 1500 and the output look-up table 1520receive n-bit input symbols via lines 1540, 1545, respectively, whilethe output of the signal mapping look-up table 1530 serves as then+1-bit output of the cyclic trellis encoder 810.

In operation, the n-bit input symbol enters the state transition look-uptable 1500. In addition, the k-bit present state of the encoder, whichis supplied by the memory element 1510, is applied to the present stateinput of the state transition look-up table 1500. Given the k-bitpresent state and the n-bit input signal, the state transition table1500 outputs a next-state value over the line 1505. The state transitionlook-up table 1500 is implemented so that the next-state value generatedby the state transition look-up table 1500 is determined in accordancewith the next-state tables of FIGS. 20A-20C.

The next-state value enters the memory element 1510 where the next-statevalue is stored for one input cycle. That is, upon application of thenext n-bit input signal, the next-state value which was applied to theinput of the memory element 1510 is passed to the output of the memoryelement 1510. Thus, the output of the memory element 1510 corresponds tothe present state of the trellis encoder 810.

The present state value at the output of the memory element 1510 isapplied to the inputs of both the state transition look-up table 1500and the encoder output look-up table 1520. The output look-up table 1520receives the present state input via the line 1525 and the input symbolvia the line 1545, and generates an encoded n+l-bit output symbol. Forapplications in AWGN, the output table 1520 is implemented so that theoutput value generated by the output look-up table 1520 is determined inaccordance with the output table of FIG. 19. For applications inRayleigh fading, the output table will be determined by the particularsignal constellation and the well known method of Gray coding.

The output value is then applied to the signal mapping look-up table1530 via the line 1535. The signal mapping look-up table 1530 assignseach of the 2^(n+1) possible output values to a point on the 2^(n+1)point signal constellation in accordance with natural or Gray coding,depending on the application.

Cyclic Trellis Decoder

As will be appreciated by one of ordinary skill in the art, a decoder(e.g., the decoder 870 of FIG. 8) for decoding data encoded by means ofthe above described encoder may be easily realized as a Viterbi decoder.Conventional Viterbi decoders decode the received data stream inaccordance with the soft decision Viterbi decoding methods described,given the form of the output look-up table 1520, the state transitionlook-up table 1500, the signal mapping look-up table 1530.

Briefly, the trellis decoder 870 includes a memory circuit indicatedgenerally as 871, which contains information from the state transitiontable 1500 concerning the trellis branches (i.e., the state transitionpaths) which merge into and out of each state. The decoder 870 alsoincludes a memory circuit indicated generally as 872, which containsinformation from the output look-up 1520 table concerning the input,output! symbol pair associated with each trellis branch. The decoder 870further includes a circuit indicated generally as 873, for calculatingthe decoding metrics which are the Euclidean distances between thereceived signals and the output signals associated with the trellisbranches, as well as a comparator circuit indicated generally as 874,for selecting the most likely path as calculated from the decodingmetrics. Of course, it should be understood that each of these circuitscould be implemented in software.

Before decoding, the received signal is converted to a digital phase andamplitude value which is fed into the decoder 870. The decoder 870 thendetermines the decoding metric associated with each state value asdetermined by the state transition and output look-up table memorycircuits. This process is repeated for many symbols, and thecorresponding metrics for each state are accumulated. A comparison ismade amongst all of the possible states as described above in thesection entitled "Maximum Likelihood Viterbi Decoding." Finally the mostlikely path or sequence of symbols is output by the decoder 870.

Catastrophic Codes

It will be appreciated by those skilled in the art that certain look-uptable encoder implementations may result in catastrophic codes. Incatastrophic codes, a finite sequence of errors in the received signalsequence may result in an indefinite sequence of decoding errors. Onesign that a code could result in a catastrophic encoder implementationis that in a next-state table, a present state transitions into the samenext state with a given output, while another present state transitionsinto the same next state (i.e., stays in the same state) with the samegiven output. Take, for example, the state transition table of FIG. 13.When the encoder is in the state 0 and a zero input is applied to theencoder, the encoder transitions back to the state 0 with an output ofzero. Similarly, when the encoder is in the present state 10 and a zeroinput is applied to the encoder, the encoder transitions back to thestate 10 with an output of zero. Thus, it is possible that such alook-up table implementation of the encoder would cause the encoder tobe catastrophic.

In order to avoid catastrophic codes, a simple modification may be madeto the state transition look-up table. For example, the table of FIG. 13could be modified as depicted in the table of FIG. 22. As shown in FIG.22, the next-state values for present state rows 10 and 14 have beenswitched so that, upon application of a zero input, the present state 10transitions into the next state 9. In this manner catastrophic codes maybe avoided with little or no significant degradation of errorperformance.

While various embodiments of the system and method of the presentinvention have been described, it should be understood that theseembodiments have been presented by way of example only, and are notintended to limit the scope of the present invention. Thus, the breadthand scope of the present invention should be defined only in accordancewith the following claims and their equivalents.

I claim:
 1. A receiver for a trellis-coded multi-level modulationcommunication system comprising:a trellis decoder which receives abaseband signal, said trellis decoder comprising:means forreconstructing a trellis structure defined by a state transition look-uptable wherein a set of present states is partitioned into subsets havingsuccessive members, and next state values assigned to each of saidpresent state subsets are shifted cyclicly for successive members ofsaid present state subset; means for determining input and outputsymbols associated with branches of said trellis structure as defined byan encoder output look-up table wherein said present states arepartitioned into two subsets and the outputs are partitioned into twosubsets so that said output look-up table outputs a symbol which belongsto a first output subset when in one of said present state subsets andoutputs a symbol which belongs to a second output subset when in theother of said present state subsets; a calculation circuit whichdetermines the Euclidean distances between points on a phase/amplitudecoordinate system corresponding to the received signals and points of aphase/amplitude signal constellation corresponding to signals associatedwith branches on said trellis structure; and comparator and selectorcircuits for selecting the most likely path of the received signal onsaid trellis structure on the basis of said determined Euclideandistances.
 2. A receiver for a trellis-coded multi-level modulationcommunication system comprising:a trellis decoder which receives abaseband signal, said trellis decoder comprising: first signalprocessing circuitry for reconstructing a trellis structure defined by astate transition input/output table wherein a set of present states ispartitioned into subsets having successive members, and next statevalues assigned to each of said present state subsets are shiftedcyclicly for successive members of said present state subset;secondsignal processing circuitry for determining input and output symbolsassociated with branches of said trellis structure as defined by anencoder output input/output table wherein said present states arepartitioned into two subsets and the outputs are partitioned into twosubsets so that said encoder output input/output table outputs a symbolwhich belongs to a first output subset when in one of said present statesubsets and outputs a symbol which belongs to a second output subsetwhen in the other of said present state subsets; a calculation circuitwhich determines distances between points on a phase/amplitudecoordinate system corresponding to the received signals and points of aphase/amplitude signal constellation corresponding to signals associatedwith branches on said trellis structure; and comparator and selectorcircuits for selecting the most likely path of the received signal onsaid trellis structure on the basis of said determined distances.
 3. Areceiver as defined in claim 2, wherein said first and second signalprocessing circuitry include a Viterbi decoder which utilizessoft-decision Viterbi decoding methods.
 4. A receiver as defined inclaim 2, further comprising a constellation signal mapper which mapssaid received signals to points on said phase/amplitude signalconstellation.
 5. A receiver for a trellis-coded multi-level modulationcommunication system comprising:a trellis decoder which receives abaseband signal, said trellis decoder comprising: a first look-up tablefor reconstructing signal constellation points divided into twosymmetrical sets of symbol points wherein said first look-up table isdefined as an inverse of an output table of output symbols wherein saidoutput table is defined to have present state rows and input symbolcolumns wherein said output symbols are determined as a function ofsymbols input to a trellis encoder and a present state of said trellisencoder, said output table being further defined by having each of saidoutput symbols assigned to said signal constellation points, said signalconstellation points partitioned into a first subset of output symbolsand a second subset of output symbols, even ones of said present staterows loaded with output symbols from said first subset, and odd ones ofsaid present state rows loaded with output symbols from said secondsubset; and a second look-up table for cyclic trellis decoding datamapped according to said signal constellation.
 6. A receiver as definedin claim 5, wherein said second look-up table is defined as an inverseof a next-state table, said next-state table defined to have presentstate rows and input symbol columns, wherein the next states are definedas a function of symbols input to said encoder and a present state ofsaid encoder, said next-state look-up table further defined tohave:first ones of said present state rows loaded with next states ofsaid encoder until at least one of said first present state rows is fulland all of said next state values have been used; and other ones of saidpresent state rows loaded with next states that are cyclicly shiftedfrom said next states in each of said first ones of said present staterows until all of said present state rows are filled.